1.6 Point groups and space groups

Definition of point groups

A point group is a mathematical group containing all symmetry operations that leave at least one point in space unmoved. Think of it this way: you grab a molecule or crystal, perform some operation on it (rotate it, reflect it, etc.), and if it looks identical afterward, that operation belongs to the point group. The "point" in the name refers to the fact that one point (usually the center of the molecule or crystal) never moves during any of these operations.

Point groups are the starting framework for classifying crystal symmetry, and they directly determine many physical properties of materials.

Symmetry operations in point groups

There are four fundamental types of symmetry operations in point groups:

• Rotation ($C_n$): Rotate the object by $360°/n$ about an axis. A $C_3$ rotation, for instance, is a 120° turn. If the object looks the same after that turn, it has three-fold rotational symmetry.
• Reflection ($\sigma$): Mirror the object across a plane. The plane can be vertical ($\sigma_v$, containing the principal axis), horizontal ($\sigma_h$, perpendicular to it), or dihedral ($\sigma_d$, bisecting two $C_2$ axes).
• Inversion ($i$): Map every point $(x, y, z)$ to $(-x, -y, -z)$ through a central point. A cube has inversion symmetry; a tetrahedron does not.
• Improper rotation ($S_n$): A rotation by $360°/n$ followed by a reflection through a plane perpendicular to the rotation axis. Note that $S_1$ is just a reflection and $S_2$ is equivalent to inversion.

Every point group is built from some combination of these four operations (plus the identity operation, which does nothing).

Classification of point groups

Crystals are restricted in which rotational symmetries they can have because their atoms must fill space periodically. This restriction (the crystallographic restriction theorem) limits rotation axes to 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold. Five-fold and higher-than-six-fold rotations are incompatible with translational periodicity.

As a result, there are exactly 32 crystallographic point groups in three dimensions. These 32 groups are distributed across the seven crystal systems:

• Triclinic (lowest symmetry, 2 point groups)
• Monoclinic (3 point groups)
• Orthorhombic (3 point groups)
• Tetragonal (7 point groups)
• Trigonal (5 point groups)
• Hexagonal (7 point groups)
• Cubic (highest symmetry, 5 point groups)

Each crystal system is defined by the minimum symmetry its point groups must possess. For example, all cubic point groups contain four three-fold axes oriented along the body diagonals of a cube.

Schoenflies notation for point groups

Schoenflies notation is widely used in molecular physics and spectroscopy. The key symbols are:

• $C_n$: cyclic group with an $n$-fold rotation axis only
• $C_{nv}$: $C_n$ plus $n$ vertical mirror planes (e.g., $C_{3v}$ describes ammonia, $\text{NH}_3$)
• $C_{nh}$: $C_n$ plus a horizontal mirror plane
• $D_n$: $C_n$ plus $n$ two-fold axes perpendicular to the principal axis
• $D_{nh}$: $D_n$ plus a horizontal mirror plane (e.g., $D_{6h}$ describes benzene)
• $D_{nd}$: $D_n$ plus dihedral mirror planes
• $S_n$: only an improper rotation axis
• $T$, $T_d$, $T_h$, $O$, $O_h$, $I$, $I_h$: cubic and icosahedral groups ($O_h$ is the full symmetry of a cube)

Hermann-Mauguin notation for point groups

Hermann-Mauguin (H-M) notation is the standard in crystallography. Instead of naming the group type, it lists the key symmetry elements along specific crystallographic directions:

• A number $n$ denotes an $n$-fold rotation axis.
• $m$ denotes a mirror plane.
• $\bar{n}$ denotes an $n$-fold rotoinversion axis (the crystallographic equivalent of improper rotation).
• A slash $/$ means "perpendicular to," so $2/m$ means a two-fold axis with a mirror plane perpendicular to it.

Examples:

• $2/m$: two-fold rotation with a perpendicular mirror (monoclinic, equivalent to Schoenflies $C_{2h}$)
• $4mm$: four-fold rotation with two sets of mirror planes containing the rotation axis (equivalent to $C_{4v}$)
• $m\bar{3}m$: full cubic symmetry (equivalent to $O_h$)

H-M notation is preferred for crystals because it directly references the crystal axes, making it easier to connect symmetry elements to the lattice geometry.

Definition of space groups

A space group describes the complete symmetry of a crystal structure, combining point group operations with translational symmetry. While a point group captures what happens around a single point, a space group accounts for the fact that a crystal is a periodic arrangement of atoms extending through space.

Every space group contains the translational symmetry of a Bravais lattice plus whatever rotations, reflections, and combined operations are compatible with that lattice.

Symmetry operations in space groups

Space groups include all point group operations, plus three types of operations that involve translation:

• Pure translation: Shift the entire crystal by a lattice vector. This is the most basic symmetry of any crystal.
• Screw axis ($n_m$): Rotate by $360°/n$ and simultaneously translate by $m/n$ of the lattice period along the rotation axis. For example, a $2_1$ screw axis rotates 180° and translates by half a lattice vector. Screw axes produce helical arrangements of atoms.
• Glide plane: Reflect across a plane and simultaneously translate by half a lattice vector parallel to that plane. Different glide types ($a$, $b$, $c$, $n$, $d$) correspond to different translation directions.

Screw axes and glide planes are the operations that distinguish space groups from simple combinations of point groups and Bravais lattices. They represent symmetries that have no counterpart in isolated molecules.

Classification of space groups

There are exactly 230 unique space groups in three dimensions. This number comes from systematically combining:

• The 7 crystal systems (which determine the shape of the unit cell)
• The 14 Bravais lattices (which add centering: primitive, body-centered, face-centered, or base-centered)
• The 32 crystallographic point groups (which supply the rotational/reflective symmetry)
• All possible screw axes and glide planes compatible with each combination

Not every combination of point group and Bravais lattice yields a distinct space group, and some combinations allow multiple space groups due to different choices of screw axes and glide planes. The full enumeration was completed independently by Fedorov, Schoenflies, and Barlow in the 1890s.

Hermann-Mauguin notation for space groups

Space group H-M symbols start with a letter indicating the lattice centering, followed by the symmetry elements:

• Lattice letter: $P$ (primitive), $I$ (body-centered), $F$ (face-centered), $A/B/C$ (base-centered), $R$ (rhombohedral)
• Symmetry elements: listed along the same crystallographic directions as the point group, but now screw axes and glide planes replace simple rotations and mirrors where appropriate

Examples:

• $P2_1/c$: primitive monoclinic lattice, $2_1$ screw axis along $b$, $c$-glide plane perpendicular to $b$. This is the single most common space group in molecular crystals.
• $Fm\bar{3}m$: face-centered cubic with full octahedral symmetry (the space group of NaCl and many FCC metals).
• $P6_3/mmc$: hexagonal with a $6_3$ screw axis, mirror planes, and a $c$-glide. This is the space group of hexagonal close-packed metals and graphite.

Number of space groups in 2D and 3D

• In two dimensions, there are 17 space groups, commonly called wallpaper groups. They describe every possible symmetry of a repeating 2D pattern. You can find all 17 in tile work, textiles, and the art of M.C. Escher.
• In three dimensions, there are 230 space groups. These cover every possible symmetry that a 3D crystal can have.

The jump from 17 to 230 reflects the much richer set of symmetry operations available in 3D (screw axes, glide planes in multiple orientations, more lattice centering options).

Relationship between point groups and space groups

Point groups and space groups are not independent classification systems. Every space group has an associated point group, and understanding how they connect is essential for working with crystal symmetry.

Point groups as subgroups of space groups

If you take a space group and strip away all translational components (pure translations, the translational parts of screw axes, and the translational parts of glide planes), what remains is the point group of the crystal. Formally, the point group is the factor group of the space group modulo its translation subgroup.

For example:

• Space group $P2_1/c$ has point group $2/m$ (the $2_1$ screw axis becomes a simple 2-fold rotation, and the $c$-glide becomes a mirror plane).
• Space group $Fd\bar{3}m$ (diamond) has point group $m\bar{3}m$ ($O_h$).

This means the macroscopic physical properties of a crystal (those that don't depend on the atomic-scale arrangement) are determined by the point group alone.

Space groups as extensions of point groups

Going the other direction, you can build a space group by starting with a point group and "decorating" it with translational elements. A single point group can give rise to multiple space groups depending on which rotations become screw axes and which mirrors become glide planes.

For example, point group $mmm$ ($D_{2h}$) generates several orthorhombic space groups, including $Pmmm$, $Pnma$, $Cmcm$, and $Fmmm$, among others. Each has the same macroscopic symmetry but different internal arrangements.

Wyckoff positions and site symmetry

Within a unit cell, not every location has the same local symmetry. Wyckoff positions are sets of points in the unit cell that are mapped onto each other by the space group operations.

Each Wyckoff position has:

• A multiplicity (how many equivalent points are in the set per unit cell)
• A site symmetry (the point group symmetry that the position retains)
• A Wyckoff letter (a label, with $a$ being the highest-symmetry position)

For example, in space group $Pm\bar{3}m$ (the perovskite structure $\text{ABO}_3$):

• The A-site cation sits at Wyckoff position $1a$ with site symmetry $m\bar{3}m$
• The B-site cation sits at $1b$ with site symmetry $m\bar{3}m$
• The oxygen atoms sit at $3c$ with lower site symmetry $4/mmm$

Wyckoff positions matter because an atom's site symmetry constrains its local environment and determines how many free parameters are needed to describe its position. Atoms on high-symmetry sites have their coordinates fully fixed by symmetry.

Applications of point groups and space groups

Symmetry in crystal structures

Space groups provide a compact way to describe an entire crystal structure. Instead of listing the coordinates of every atom in the crystal, you specify:

1. The space group
2. The lattice parameters ($a, b, c, \alpha, \beta, \gamma$)
3. The atoms in the asymmetric unit (the smallest portion of the unit cell from which the full structure can be generated by applying all space group operations)

Some well-known structures and their space groups:

• Diamond (C, Si, Ge): $Fd\bar{3}m$ (space group #227)
• Graphite: $P6_3/mmc$ (space group #194)
• Perovskite ($\text{BaTiO}_3$ cubic phase): $Pm\bar{3}m$ (space group #221)
• Rock salt (NaCl): $Fm\bar{3}m$ (space group #225)

Symmetry-based selection rules for transitions

Point group symmetry determines which transitions between quantum states are allowed or forbidden. The underlying principle is that a transition matrix element is nonzero only if the direct product of the representations of the initial state, the operator, and the final state contains the totally symmetric representation.

Two important selection rules:

• Laporte rule: In molecules or ions with an inversion center (centrosymmetric point groups), electric dipole transitions between states of the same parity (both gerade or both ungerade) are forbidden. This is why d-d transitions in octahedral transition metal complexes are weak.
• Spin selection rule: Transitions between states of different spin multiplicity are forbidden in the absence of spin-orbit coupling.

Symmetry in phonon dispersion relations

The symmetry of a crystal constrains its vibrational (phonon) spectrum. At high-symmetry points in the Brillouin zone, phonon modes are classified by the irreducible representations of the little group (the point group of the wavevector).

This classification tells you:

• Which modes are degenerate (have the same frequency due to symmetry)
• Where band crossings or anti-crossings can occur
• Which modes are Raman-active, infrared-active, or silent (determined by the symmetry of the mode)

Key high-symmetry points in the Brillouin zone include $\Gamma$ (zone center, $\mathbf{k} = 0$), $X$ (zone boundary along a cube axis), and $L$ (zone boundary along a body diagonal) for cubic crystals.

Symmetry in electronic band structures

The electronic band structure of a crystal is also governed by space group symmetry. Bloch states at a given $\mathbf{k}$-point transform according to the irreducible representations of the little group of $\mathbf{k}$.

Consequences include:

• Band degeneracies at high-symmetry points (e.g., the valence band top in diamond-structure semiconductors is triply degenerate at $\Gamma$ before spin-orbit coupling splits it)
• Compatibility relations that dictate how bands connect between different high-symmetry points
• Band gaps whose character (direct vs. indirect) is related to symmetry. GaAs (space group $F\bar{4}3m$) has a direct gap at $\Gamma$, while Si (space group $Fd\bar{3}m$) has an indirect gap with the conduction band minimum near $X$.

The higher symmetry of Si's diamond structure (which includes inversion) versus GaAs's zinc blende structure (which lacks inversion) has direct consequences for optical properties: GaAs is a much more efficient light emitter.

Determination of point groups and space groups

Experimental methods for symmetry determination

Several experimental techniques probe crystal symmetry:

• Diffraction methods (X-ray, neutron, electron): These are the primary tools. They reveal the lattice parameters, crystal system, and space group through the geometry and intensities of diffraction spots.
• Spectroscopy (infrared and Raman): The number and activity of vibrational modes depend on the point group. Comparing observed IR and Raman spectra with group theory predictions helps confirm or narrow down the point group.
• Polarized light microscopy: Optical properties like birefringence depend on the crystal system. Uniaxial crystals (tetragonal, hexagonal, trigonal) and biaxial crystals (orthorhombic, monoclinic, triclinic) can be distinguished, and cubic crystals are optically isotropic.

X-ray diffraction and space group determination

X-ray diffraction is the workhorse technique for determining space groups. The process works roughly as follows:

1. Collect the diffraction pattern and index the reflections to determine the lattice parameters and crystal system.

2. Check the Laue symmetry of the diffraction pattern to narrow down the point group. (Diffraction patterns always have inversion symmetry due to Friedel's law, so you determine the Laue class, not the full point group directly.)

3. Look for systematic absences. Certain classes of reflections will be missing if screw axes or glide planes are present. For example:

   • A $2_1$ screw axis along $b$ causes $0k0$ reflections to be absent when $k$ is odd
   • A $c$-glide plane perpendicular to $b$ causes $h0l$ reflections to be absent when $l$ is odd

4. Combine the Laue class and systematic absences to identify the space group. In some cases, ambiguities remain and must be resolved by structure refinement.

Computational methods for symmetry analysis

Computational tools are essential for practical symmetry work:

• Group theory software can determine the point group from a set of atomic coordinates, find Wyckoff positions, and generate character tables and irreducible representations.
• First-principles calculations (e.g., density functional theory) predict equilibrium structures, and the symmetry of the relaxed structure can be analyzed automatically.
• Widely used tools include FINDSYM (identifies the space group from atomic coordinates with adjustable tolerance), the Bilbao Crystallographic Server (provides Wyckoff positions, irreducible representations, and compatibility relations), and ISOTROPY (analyzes symmetry-lowering phase transitions).

Consequences of symmetry breaking

When a crystal undergoes a phase transition that lowers its symmetry, new physical properties can emerge that were forbidden by the original, higher-symmetry structure. The relationship between the high-symmetry and low-symmetry phases is described by group-subgroup relations: the space group of the low-symmetry phase is a subgroup of the high-symmetry phase's space group.

Ferroelectricity and noncentrosymmetric point groups

A material can only be ferroelectric (possess a switchable spontaneous electric polarization) if its point group is both polar (has a unique polar axis) and noncentrosymmetric (lacks inversion symmetry). Of the 32 crystallographic point groups, 10 are polar.

Barium titanate ($\text{BaTiO}_3$) is a classic example. Above 120°C, it has the cubic perovskite structure (point group $m\bar{3}m$), which is centrosymmetric and nonpolar. Below 120°C, the Ti ion shifts off-center, lowering the symmetry to tetragonal point group $4mm$, which is polar. This symmetry-breaking transition creates the spontaneous polarization.

Piezoelectricity and noncentrosymmetric point groups

Piezoelectricity (generating electric charge under mechanical stress) requires only that the point group lack an inversion center. Of the 32 point groups, 21 are noncentrosymmetric, and 20 of those are piezoelectric (the cubic point group $432$ is the exception because its high symmetry causes the piezoelectric tensor components to cancel).

• Quartz ($\text{SiO}_2$, point group $32$): used in oscillators and frequency standards
• Zinc oxide (ZnO, point group $6mm$): used in sensors and actuators

Note that all ferroelectric materials are also piezoelectric (since polar groups are a subset of noncentrosymmetric groups), but not all piezoelectrics are ferroelectric.

Magnetism and time-reversal symmetry breaking

Magnetic order breaks time-reversal symmetry: reversing the direction of time would reverse all currents and spins, flipping the magnetization. Ordinary (non-magnetic) point groups and space groups don't account for this, so magnetic structures require an extended framework.

Magnetic point groups (also called Shubnikov groups) include the time-reversal operation as an additional symmetry element. There are 122 magnetic point groups (compared to 32 ordinary ones) and 1651 magnetic space groups (compared to 230).

Examples of magnetic materials and their crystallographic (non-magnetic) space groups:

• Iron (BCC, $Im\bar{3}m$): ferromagnetic below 770°C
• Nickel (FCC, $Fm\bar{3}m$): ferromagnetic below 358°C

The magnetic space group provides additional information about how the spin arrangement relates to the crystal structure, which is especially important for antiferromagnets and other complex magnetic orders.